Fokker–Planck equation

A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. In this case the initial condition is a Dirac delta function centered away from zero velocity. Over time the distribution widens due to random impulses.

The transition probability, Pt,t′(x|x′){\displaystyle \mathbb {P} _{t,t'}(x|x')}, the probability of going from (t′,x′){\displaystyle (t',x')} to (t,x){\displaystyle (t,x)}, is introduced here; the expectation can be written as

then we arrive to the Kolmogorov Forward Equation, or Fokker-Planck Equation which, simplifying the notation p(x,t)=Pt,t′(x|x′){\displaystyle p(x,t)=\mathbb {P} _{t,t'}(x|x')}, in its differential form reads

While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman-Kac formula can be used, which is a consequence of the Kolmogorov backward equation.

The stochastic process defined above in the Itō sense can be rewritten within the Stratonovich convention as a Stratonovich SDE:

It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itō SDE.

The zero drift equation with constant diffusion can be considered as a model of classical Brownian motion:

In plasma physics, the distribution function for a particle species s{\displaystyle s}, ps(x→,v→,t){\displaystyle p_{s}\left({\vec {x}},{\vec {v}},t\right)}, takes the place of the probability density function. The corresponding Boltzmann equation is given by

where the third term includes the particle acceleration due to the Lorentz force and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities ⟨Δvi⟩{\displaystyle \langle \Delta v_{i}\rangle } and ⟨ΔviΔvj⟩{\displaystyle \langle \Delta v_{i}\Delta v_{j}\rangle } are the average change in velocity a particle of type s{\displaystyle s} experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere.[10] If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.

Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (the Monte Carlo method, canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider the probability f(v,t)dv{\displaystyle f(\mathbf {v} ,t)d\mathbf {v} } of the particle having a velocity in the interval (v,v+dv){\displaystyle (\mathbf {v} ,\mathbf {v} +d\mathbf {v} )} when it starts its motion with v0{\displaystyle \mathbf {v} _{0}} at time 0.

Being a partial differential equation, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker–Planck equation with the Schrödinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. In many applications, one is only interested in the steady-state probability distribution f0(x){\displaystyle f_{0}(x)}, which can be found from f˙0(x)=0{\displaystyle {\dot {f}}_{0}(x)=0}. The computation of mean first passage times and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.

In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient σ(Xt,t){\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker–Planck equation: Given the density f(x,t) of the option underlying X deduced from the option market, one aims at finding the local volatility σ(Xt,t){\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution. Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility σ(Xt,t){\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a solution of the Fokker–Planck equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008) and Musiela and Rutkowski (2008).

Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods.[11] This is used, for instance, in critical dynamics.

A derivation of the path integral is possible in the same way as in quantum mechanics, simply because the Fokker–Planck equation is formally equivalent to the Schrödinger equation. Here are the steps for a Fokker–Planck equation with one variable x. Write the FP equation in the form

The variables x~{\displaystyle {\tilde {x}}} conjugate to x{\displaystyle x} are called "response variables".[12]

Although formally equivalent, different problems may be solved more easily in the Fokker–Planck equation or the path integral formulation. The equilibrium distribution for instance may be obtained more directly from the Fokker–Planck equation.